3.21.66 \(\int \frac {b+a x^6}{x^6 (-b+a x^3) \sqrt [4]{b x+a x^4}} \, dx\) [2066]

Optimal. Leaf size=149 \[ \frac {4 \left (b+a x^3\right ) \left (b x+a x^4\right )^{3/4}}{21 b^2 x^6}-\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2}-\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2} \]

[Out]

4/21*(a*x^3+b)*(a*x^4+b*x)^(3/4)/b^2/x^6-1/3*2^(3/4)*(a^(7/4)+a^(3/4)*b)*arctan(2^(1/4)*a^(1/4)*(a*x^4+b*x)^(3
/4)/(a*x^3+b))/b^2-1/3*2^(3/4)*(a^(7/4)+a^(3/4)*b)*arctanh(2^(1/4)*a^(1/4)*(a*x^4+b*x)^(3/4)/(a*x^3+b))/b^2

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Rubi [A]
time = 0.77, antiderivative size = 253, normalized size of antiderivative = 1.70, number of steps used = 14, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2081, 6857, 277, 270, 477, 476, 508, 472, 218, 212, 209} \begin {gather*} -\frac {2^{3/4} a^{3/4} \sqrt [4]{x} (a+b) \sqrt [4]{a x^3+b} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 b^2 \sqrt [4]{a x^4+b x}}-\frac {2^{3/4} a^{3/4} \sqrt [4]{x} (a+b) \sqrt [4]{a x^3+b} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 b^2 \sqrt [4]{a x^4+b x}}+\frac {4 (a+b) \left (a x^3+b\right )^2}{21 a b^2 x^5 \sqrt [4]{a x^4+b x}}-\frac {4 \left (a x^3+b\right )}{21 a x^5 \sqrt [4]{a x^4+b x}}-\frac {4 \left (a x^3+b\right )}{21 b x^2 \sqrt [4]{a x^4+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + a*x^6)/(x^6*(-b + a*x^3)*(b*x + a*x^4)^(1/4)),x]

[Out]

(-4*(b + a*x^3))/(21*a*x^5*(b*x + a*x^4)^(1/4)) - (4*(b + a*x^3))/(21*b*x^2*(b*x + a*x^4)^(1/4)) + (4*(a + b)*
(b + a*x^3)^2)/(21*a*b^2*x^5*(b*x + a*x^4)^(1/4)) - (2^(3/4)*a^(3/4)*(a + b)*x^(1/4)*(b + a*x^3)^(1/4)*ArcTan[
(2^(1/4)*a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)])/(3*b^2*(b*x + a*x^4)^(1/4)) - (2^(3/4)*a^(3/4)*(a + b)*x^(1/4)*(
b + a*x^3)^(1/4)*ArcTanh[(2^(1/4)*a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)])/(3*b^2*(b*x + a*x^4)^(1/4))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 476

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 508

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[k*(a^(p + (m + 1)/n)/n), Subst[Int[x^(k*((m + 1)/n) - 1)*((c - (b*c - a*d)*x^k)^q/(1 - b*x^k)^(p +
q + (m + 1)/n + 1)), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {b+a x^6}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {b+a x^6}{x^{25/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \left (\frac {b}{a x^{25/4} \sqrt [4]{b+a x^3}}+\frac {1}{x^{13/4} \sqrt [4]{b+a x^3}}+\frac {a b+b^2}{a x^{25/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}}\right ) \, dx}{\sqrt [4]{b x+a x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}}+\frac {\left (b \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{25/4} \sqrt [4]{b+a x^3}} \, dx}{a \sqrt [4]{b x+a x^4}}+\frac {\left (b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{25/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}} \, dx}{a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{9 b x^2 \sqrt [4]{b x+a x^4}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{b+a x^3}} \, dx}{7 \sqrt [4]{b x+a x^4}}+\frac {\left (4 b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{x^{22} \left (-b+a x^{12}\right ) \sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {\left (1-a x^4\right )^2}{x^8 \left (-b+2 a b x^4\right )} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 a b \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \left (-\frac {1}{b x^8}+\frac {a^2}{b \left (-1+2 a x^4\right )}\right ) \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 a b \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{b x+a x^4}}+\frac {\left (4 a (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{-1+2 a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{b x+a x^4}}-\frac {\left (2 a (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}}-\frac {\left (2 a (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{21 a x^5 \sqrt [4]{b x+a x^4}}-\frac {4 \left (b+a x^3\right )}{21 b x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{b x+a x^4}}-\frac {2^{3/4} a^{3/4} (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}}-\frac {2^{3/4} a^{3/4} (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b^2 \sqrt [4]{b x+a x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 15.16, size = 97, normalized size = 0.65 \begin {gather*} \frac {4 \left (\left (b+a x^3\right )^2-\frac {7 a (a+b) x^6 \sqrt [4]{1+\frac {a x^3}{b}} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {2 a x^3}{b-a x^3}\right )}{\sqrt [4]{1-\frac {a x^3}{b}}}\right )}{21 b^2 x^5 \sqrt [4]{x \left (b+a x^3\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(b + a*x^6)/(x^6*(-b + a*x^3)*(b*x + a*x^4)^(1/4)),x]

[Out]

(4*((b + a*x^3)^2 - (7*a*(a + b)*x^6*(1 + (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (-2*a*x^3)/(b - a*
x^3)])/(1 - (a*x^3)/b)^(1/4)))/(21*b^2*x^5*(x*(b + a*x^3))^(1/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{6}+b}{x^{6} \left (a \,x^{3}-b \right ) \left (a \,x^{4}+b x \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^6+b)/x^6/(a*x^3-b)/(a*x^4+b*x)^(1/4),x)

[Out]

int((a*x^6+b)/x^6/(a*x^3-b)/(a*x^4+b*x)^(1/4),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/x^6/(a*x^3-b)/(a*x^4+b*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^6 + b)/((a*x^4 + b*x)^(1/4)*(a*x^3 - b)*x^6), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/x^6/(a*x^3-b)/(a*x^4+b*x)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{6} + b}{x^{6} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} - b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**6+b)/x**6/(a*x**3-b)/(a*x**4+b*x)**(1/4),x)

[Out]

Integral((a*x**6 + b)/(x**6*(x*(a*x**3 + b))**(1/4)*(a*x**3 - b)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (117) = 234\).
time = 0.41, size = 272, normalized size = 1.83 \begin {gather*} -\frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, b^{2}} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, b^{2}} + \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{12 \, b^{2}} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{12 \, b^{2}} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{4}}}{21 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/x^6/(a*x^3-b)/(a*x^4+b*x)^(1/4),x, algorithm="giac")

[Out]

-1/6*sqrt(2)*(2^(3/4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/x
^3)^(1/4))/(-a)^(1/4))/b^2 - 1/6*sqrt(2)*(2^(3/4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*arctan(-1/2*2^(1/4)*(2^
(3/4)*(-a)^(1/4) - 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/b^2 + 1/12*sqrt(2)*(2^(3/4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3
/4)*b)*log(2^(3/4)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/b^2 - 1/12*sqrt(2)*(2^(3
/4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*log(-2^(3/4)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a
+ b/x^3))/b^2 + 4/21*(a + b/x^3)^(7/4)/b^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {a\,x^6+b}{x^6\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b + a*x^6)/(x^6*(b*x + a*x^4)^(1/4)*(b - a*x^3)),x)

[Out]

-int((b + a*x^6)/(x^6*(b*x + a*x^4)^(1/4)*(b - a*x^3)), x)

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